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Synchronous Machine Round Rotor (fundamental)

Round-rotor synchronous machine with fundamental parameterization

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Machines / Synchronous Machine (Round Rotor)

Description

The Synchronous Machine Round Rotor (fundamental) block models a round-rotor synchronous machine with parameterization using fundamental parameters.

Electrical Defining Equations

The synchronous machine equations are expressed with respect to a rotating reference frame defined by the equation

${\theta }_{e}\left(t\right)=N*{\theta }_{r}\left(t\right),$

where:

• θe is the electrical angle.

• N is the number of pole pairs.

• θr is the rotor angle.

Park's transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. Park's transformation is defined by

${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$

Park's transformation is used to define the per-unit synchronous machine equations. The stator voltage equations are defined by

${e}_{d}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{d}}{\text{d}t}-{\Psi }_{q}{\omega }_{r}-{R}_{a}{i}_{d},$

${e}_{q}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{q}}{\text{d}t}+{\Psi }_{d}{\omega }_{r}-{R}_{a}{i}_{q},$

and

${e}_{0}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{0}}{dt}-{R}_{a}{i}_{0},$

where:

• ed, eq, and e0 are the d-axis, q-axis, and zero-sequence stator voltages, defined by

$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right],$

where va, vb, and vc are the stator voltages measured from port ~ to neutral port n.

• ωbase is the per-unit base electrical speed.

• ψd, ψq, and ψ0 are the d-axis, q-axis, and zero-sequence stator flux linkages.

• ωr is the per-unit rotor rotational speed.

• Ra is the stator resistance.

• id, iq and i0 are the d-axis, q-axis, and zero-sequence stator currents, defined by

$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right],$

where ia, ib, and ic are the stator currents flowing from port ~ to port n.

The rotor voltage equations are defined by

${e}_{fd}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{fd}}{dt}+{R}_{fd}{i}_{fd},$

${e}_{1d}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{1d}}{dt}+{R}_{1d}{i}_{1d}=0,$

${e}_{1}{}_{q}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{1q}}{dt}+{R}_{1q}{i}_{1q}=0,$

and

${e}_{2}{}_{q}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{2q}}{dt}+{R}_{2q}{i}_{2q}=0,$

where:

• efd is the field voltage.

• e1d, e1q, and e2q are the voltages across the d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2. They are all equal to 0.

• ψfd, ψ1d, ψ1q, and ψ2q are the magnetic fluxes linking the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

• Rfd, R1d, R1q, and R2q are the resistances of rotor field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

• ifd, i1d, i1q, and i2q are the currents flowing in the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

The saturation equations are defined by

${\psi }_{at}=\sqrt{{\psi }_{d}^{2}+{\psi }_{q}^{2}},$

${K}_{s}=1$ (If saturation is disabled),

${K}_{s}=f\left({\psi }_{at}\right)$ (If saturation is enabled),

${L}_{ad}={K}_{s}*{L}_{adu},$

and

${L}_{aq}={K}_{s}*{L}_{aqu},$

where:

• ψat is the air-gap flux linkage.

• Ks is the saturation factor.

• Ladu is the unsaturated mutual inductance of the stator d-axis.

• Lad is the mutual inductance of the stator d-axis.

• Laqu is the unsaturated mutual inductance of the stator q-axis.

• Laq is the mutual inductance of the stator q-axis.

The saturation factor function, f, is calculated from the per-unit open-circuit lookup table as:

${L}_{ad}=\frac{d{\psi }_{at}}{d{i}_{fd}},$

${V}_{ag}=g\left({i}_{fd}\right),$

and

${L}_{ad}=\frac{dg\left({i}_{fd}\right)}{d{i}_{fd}}=\frac{d{V}_{ag}}{d{i}_{fd}},$

where:

• Vag is the per-unit air-gap voltage.

In per-unit,

${K}_{s}=\frac{{L}_{ad}}{{L}_{adu}},$

and

${\psi }_{at}={V}_{ag}$

can be rearranged to

${K}_{s}=f\left({\psi }_{at}\right).$

The stator flux linkage equations are defined by

${\Psi }_{d}=-\left({L}_{ad}+{L}_{i}\right){i}_{d}\text{​}+{L}_{ad}{i}_{fd}+{L}_{ad}{i}_{1d},$

$\Psi q=-\left({L}_{aq}+{L}_{i}\right){i}_{q}\text{​}+{L}_{aq}{i}_{1q}+{L}_{aq}{i}_{2q},$

and

${\Psi }_{0}=-{L}_{0}{i}_{0},$

where:

• Ll is the stator leakage inductance.

• Lad and Laq are the mutual inductances of the stator d-axis and q-axis.

The rotor flux linkage equations are defined by

${\psi }_{fd}={L}_{ffd}{i}_{fd}+{L}_{f1d}{i}_{1d}-{L}_{ad}{i}_{d},$

${\psi }_{1d}={L}_{f1d}{i}_{fd}+{L}_{11d}{i}_{1d}-{L}_{ad}{i}_{d},$

${\psi }_{1q}={L}_{11q}{i}_{1q}+{L}_{aq}{i}_{2q}-{L}_{aq}{i}_{q},$

and

${\psi }_{2q}={L}_{aq}{i}_{1q}+{L}_{22q}{i}_{2q}-{L}_{aq}{i}_{q},$

where:

• Lffd, L11d, L11q, and L22q are the self-inductances of the rotor field circuit, d-axis damper winding 1, q-axis damper winding 1 and q-axis damper winding 2. Lf1d is the rotor field circuit and d-axis damper winding 1 mutual inductance. They are defined by the following equations.

• ${L}_{ffd}={L}_{ad}+{L}_{fd}$

• ${L}_{f1d}={L}_{ffd}-{L}_{fd}$

• ${L}_{11d}={L}_{f1d}+{L}_{1d}$

• ${L}_{11q}={L}_{aq}+{L}_{1q}$

• ${L}_{22q}={L}_{aq}+{L}_{2q}$

These equations assume that per-unit mutual inductance L12q = Laq, i.e., the stator and rotor currents in the q-axis all link a single mutual flux represented by Laq.

The rotor torque is defined by

${T}_{e}={\Psi }_{d}{i}_{q}-{\Psi }_{q}{i}_{d}.$

Display Options

For synchronous machine blocks, you can perform display actions using the Power Systems menu on the block context menu.

Right-click the block. From the context menu, select one of the following from the Power Systems > Synchronous Machine menu:

• Display Base Values displays the machine per-unit base values at the MATLAB® command prompt.

• Display Associated Base Values displays associated per-unit base values at the MATLAB command prompt.

• Associated Initial Conditions displays associated initial conditions at the MATLAB command prompt.

• Plot Open-Circuit Saturation (pu) plots air-gap voltage, Vag, versus field current, ifd, (both measured in per-unit) in a MATLAB figure window. The plot contains three traces:

• Unsaturated: Stator d-axis mutual inductance (unsaturated), Ladu you specify

• Saturated: Per-unit open-circuit lookup table (Vag versus ifd) you specify

• Derived: Open-circuit lookup table (per-unit) derived from the Per-unit open-circuit lookup table (Vag versus ifd) you specify. This data is used to calculate the saturation factor, Ks, versus magnetic flux linkage, ψat, characteristic.

• Plot Saturation Factor (pu) plots saturation factor, Ks, versus magnetic flux linkage, ψat, (both measured in per-unit) in a MATLAB figure window using the present machine parameters. This is derived from parameters you specify:

• Stator d-axis mutual inductance (unsaturated), Ladu

• Per-unit field current saturation data, ifd

• Per-unit air-gap voltage saturation data, Vag

Dialog Box and Parameters

Main Tab

Rated apparent power

Rated apparent power. The default value is 555e6 VA.

Rated voltage

RMS rated line-line voltage. The default value is 24e3 V.

Rated electrical frequency

Nominal electrical frequency at which rated apparent power is quoted. The default value is 60 Hz.

Number of pole pairs

Number of machine pole pairs. The default value is 1.

Specify field circuit input required to produce rated terminal voltage at no load by

Select between

• Field circuit voltage

• Field circuit current

The default value is Field circuit current.

Field circuit current

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to Field circuit current. The default value is 1300 A.

Field circuit voltage

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to Field circuit voltage. The default value is 92.95 V.

Impedances Tab

Stator d-axis mutual inductance (unsaturated), Ladu

Unsaturated stator d-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator d-axis mutual inductance. The default value is 1.66 pu.

Stator q-axis mutual inductance (unsaturated), Laqu

Unsaturated stator q-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator q-axis mutual inductance. The default value is 1.61 pu.

Stator zero-sequence inductance, L0

Stator zero-sequence inductance. The default value is 0 pu.

Stator leakage inductance, Ll

Stator leakage inductance. The default value is 0.15 pu.

Stator resistance, Ra

Stator resistance. The default value is 0.003 pu.

Rotor field circuit inductance, Lfd

Rotor field circuit inductance. The default value is 0.165 pu.

Rotor field circuit resistance, Rfd

Rotor field circuit resistance. The default value is 0.0006 pu.

Rotor d-axis damper winding 1 inductance, L1d

Rotor d-axis damper winding 1 inductance. The default value is 0.1713 pu.

Rotor d-axis damper winding 1 resistance, R1d

Rotor d-axis damper winding 1 resistance. The default value is 0.0284 pu.

Rotor q-axis damper winding 1 inductance, L1q

Rotor q-axis damper winding 1 inductance. The default value is 0.7252 pu.

Rotor q-axis damper winding 1 resistance, R1q

Rotor q-axis damper winding 1 resistance. The default value is 0.00619 pu.

Rotor q-axis damper winding 2 inductance, L2q

Rotor q-axis damper winding 2 inductance. The default value is 0.125 pu.

Rotor q-axis damper winding 2 resistance, R2q

Rotor q-axis damper winding 2 resistance. The default value is 0.02368 pu.

Saturation Tab

Magnetic saturation representation

Block magnetic saturation representation. Options are:

• None

• Per-unit open-circuit lookup table (Vag versus ifd)

The default value is None.

Per-unit field current saturation data, ifd

The field current, ifd, data populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter is only visible when you set Magnetic saturation representation to Per-unit open-circuit lookup table (Vag versus ifd). This parameter must contain a vector with at least five elements. The default value is [0.00, 0.48, 0.76, 1.38, 1.79] pu.

Per-unit air-gap voltage saturation data, Vag

The air-gap voltage, Vag, data populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter is only visible when you set Magnetic saturation representation to Per-unit open-circuit lookup table (Vag versus ifd). This parameter must contain a vector with at least five elements. The default value is [0.00, 0.80, 1.08, 1.31, 1.40] pu.

Initial Conditions Tab

Specify initialization by

Options include:

• Electrical power and voltage output

• Mechanical and magnetic states

The default value is Electrical power and voltage output.

Terminal voltage magnitude

Initial RMS line-line voltage. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 24e3 V.

Terminal voltage angle

Initial voltage angle. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 0 deg.

Terminal active power

Initial active power. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 500e6 VA.

Terminal reactive power

Initial reactive power. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 0 VA.

Initial rotor angle

Initial rotor angle. During steady-state operation, set this parameter to the sum of the load angle and required terminal voltage offset. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 deg.

Initial stator d-axis magnetic flux linkage

Stator d-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial stator q-axis magnetic flux linkage

Stator q-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial stator zero-sequence magnetic flux linkage

Zero-sequence initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial field circuit magnetic flux linkage

Field circuit initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial d-axis damper winding 1 magnetic flux linkage

d-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial q-axis damper winding 1 magnetic flux linkage

The q-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Initial q-axis damper winding 2 magnetic flux linkage

The q-axis damper winding 2 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.

Ports

The block has the following ports:

fd+

Electrical conserving port corresponding to the field winding positive terminal

fd-

Electrical conserving port corresponding to the field winding negative terminal

R

Mechanical rotational conserving port associated with the machine rotor

C

Mechanical rotational conserving port associated with the machine case

pu

Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

• pu_fd_Efd

• pu_fd_Ifd

• pu_torque

• pu_velocity

• pu_ed

• pu_ed

• pu_e0

• pu_id

• pu_iq

• pu_i0

~

Expandable three-phase port associated with the stator windings

n

Electrical conserving port associated with the neutral point of the wye winding configuration

References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Lyshevski, S. E. Electromechanical Systems, Electric Machines and Applied Mechatronics. Boca Raton, FL: CRC Press, 1999.